This paper is devoted to some application of nonsmooth critical point theory to elasticity. In the recent years much interest has been paid to critical points for nonsmooth functionals, especially by the first author and his coworkers. In this paper the authors apply the techniques of modern nonsmooth critical points to nonlinear elasticity, a field where variational problems described by nonsmooth functionals often appear. In particular, the authors study the buckling of shearable nonlinearly elastic rods in the presence of rigid obstacles. The assumptions of nonsmooth critical points hold for this problem, and it is proved that the ``nonsmooth critical points'' satisfy Euler-Lagrange conditions which correspond to the physical equilibrium conditions. The paper contains many sections. After a very interesting historical introduction to critical point theory and nonlinear elasticity, the authors present the basic results on nonsmooth critical points. In the second part of the paper, the rod theory due essentially to Antman and Schuricht is presented. Moreover, the equations are given which govern planar equilibrium configurations of nonlinearly elastic rods. Finally, the problem of buckling of rods is studied. Both the cases of free rods and rods in contact with obstacles are considered. The total energy of a free rod is represented by a functional defined on a suitable Finsler manifold, and the critical points of the total energy represent buckling states of the free rod. Such critical points of total energy are fully studied. Under some constitutive assumptions, the authors prove existence and multiplicity results for critical points, providing that buckling can occur. Moreover, it is proved that such ``nonsmooth critical points'' defined by means of weak slope are ``physically reasonable'', since they satisfy the Euler-Lagrange equations associated with the total energy. An analogous analysis is carried out also for buckling of constrained nonlinear rods.